Question

Cans of coke use the data and the claim given in exercise 1 to identify the null and alternative hypothesis and the test statistic. What is the sampling distribution of the test statistic?

Short answer

The hypotheses areas follows:

$$\begin{gathered} H_o:\sigma =0.115 \\ {H}_1:\sigma \ne 0.115 \end{gathered}$$

The test statistic would follow chi-square distribution and has a value of 49.024 with 9 degrees of freedom.

Step-by-step solution

Step-1: Given information

Refer to example 1 for summary statistics.

$n=10,\overline{x}=12.0004oz,s=0.115oz$

Claim states that the standard deviation for the population is equal to 0.115 oz.

Step-2: State the hypothesis

Let $\sigma$be the actual standard deviation of the population.

The hypotheses are formulated asfollows:

$$\begin{gathered} H_o:\sigma =0.115 \\ {H}_1:\sigma \ne 0.115 \end{gathered}$$

Step-3: Describe the test statistic

The test statistic for the claim would follow the chi-square distribution with n-1 degrees of freedom.

The formula forthetest statistic isas follows:

$$\begin{gathered} {\mathbf{\chi }}^{\mathbf{2}}\mathbf{=}\frac{\left(\mathbf{n}\mathbf{-}\mathbf{1}\right){\mathbf{s}}^{\mathbf{2}}}{{\mathbf{\sigma }}^{\mathbf{2}}} \\ =\frac{\left(10-1\right){\left(0.2684\right)}^2}{{\left(0.115\right)}^2} \\ =49.024 \end{gathered}$$

The degree of freedom isas follows:

$$\begin{gathered} df=n-1 \\ =10-1 \\ =9 \end{gathered}$$

Thus, the test statistic value is 49.024 with 9 degrees of freedom.